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What's the deal with Euler's identity?

39 points by surprisetalk

by ginkoleaf

0 subcomment

I do not think I appreciated the formula until I had been exposed to the exponential map in its generalized forms: The so-called 'matrix exponential' and the exponential map of Lie algebra. They place Euler's humble formula into a grander and rather beautiful setting.
Now, I like to think of exponentiation as a kind of integral over infinitesimal generators; and $i$ just happens to be a generator for rotation about a circle in the plane (aka $\mathrm{U}(1)$ aka $x \mapsto e^{it}x$).

by rmunn

6 subcomments

Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi:
e^(i*tau) = 1
I won't reproduce https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).

by zkmon

2 subcomments

by ccppurcell

0 subcomment

It's obviously of more interest as a piece of outreach than as a piece of mathematics. Nevertheless I've always wondered about the e^ipi + 1 = 0 formulation. It seems ugly and ad hoc, and the connection between the "5 constants" is not all that meaningful.
That e^ipi = -1 is related to the much more profound observation that the complex numbers represent a sort of rotation into a previously unknown dimension of numbers.

by stared

0 subcomment

Here is the Euler's identity in my recent side project, equations visualised - https://p.migdal.pl/equations-explained-colorfully/#euler.

by aap_

0 subcomment

Nobody ever considers the spinorial version. e^iπ is a 360° rotation on a spinor, and + is averaging spinors rotationally. so e^iπ + 1 = 0 means there is no way to interpolate between the identity and a twist in the spinor, because the axis of a 360° rotation is undefined.
Things get so much more fun once you embrace spinors.

by sevensor

0 subcomment

I do like this explanation better. I learned the Maclaurin series explanation in school, where you can show that the series approximations line up, but I never felt that explained why it worked. The idea of starting with -1 as a half rotation and then taking fractions of that really appeals to the intuition.

by xeonmc

2 subcomments

Never liked that form of the Euler's formula. I prefer the following:
```
    (-1)ˣ = cos(πx) + i sin(πx)
```

by constantcrying

0 subcomment

The real magic is that complex exponentials are describing rotations in R^2.
To be honest, this equation completely fails to represent this.